Improved upper bounds for vertex and edge fault diameters of Cartesian graph bundles

Mixed fault diameter of a graph $G$, $ \D_{(a,b)}(G)$, is the maximal diameter of $G$ after deletion of any $a$ vertices and any $b$ edges. Special cases are the (vertex) fault diameter $\D^V_{a} = \D_{(a,0)}$ and the edge fault diameter $\D^E_{a} = \D_{(0,a)}$. Let $G$ be a Cartesian graph bundle with fibre $F$ over the base graph $B$. We show that (1) $\D^V_{a+b+1}(G)\leq \D^V_{a}(F)+\D^V_{b}(B)$ when the graphs $F$ and $B$ are $k_F$-connected and $k_B$-connected, $0< a < k_F$, $0< b < k_B$, and provided that $\D_{(a-1,1)}(F)\leq \D^{V}{a} (F)$ and $\D{(b-1,1)}(B)\leq \D^{V}{b} (B)$ and (2) $\D^E{a+b+1}(G)\leq \D^E_{a}(F)+\D^E_{b}(B)$ when the graphs $F$ and $B$ are $k_F$-edge connected and $k_B$-edge connected, $0\leq a < k_F$, $0\leq b < k_B$, and provided that $\D^E_{a}(F)\geq 2$ and $\D^E_{b}(B)\geq 2$.

💡 Research Summary

The paper investigates the fault‑tolerant diameter of Cartesian graph bundles, a class of networks obtained by “bundling’’ a fibre graph (F) over a base graph (B). In fault‑tolerant network design, the fault diameter measures the worst‑case distance between two surviving vertices after a certain number of components (vertices or edges) have failed. The authors introduce the mixed fault diameter (\Delta_{(a,b)}(G)), defined as the maximum diameter of a graph (G) after the deletion of any (a) vertices and any (b) edges. The special cases (\Delta^{V}{a}=\Delta{(a,0)}) (vertex fault diameter) and (\Delta^{E}{a}=\Delta{(0,a)}) (edge fault diameter) are the focus of most previous work.

A Cartesian graph bundle (G) can be visualised as follows: for each vertex (v) of the base graph (B) there is a copy (F_{v}) of the fibre graph (F); for each edge ((u,v)) of (B) a set of “transition’’ edges connects the corresponding copies (F_{u}) and (F_{v}). This construction generalises many well‑known topologies such as hypercubes, tori, and grid networks, making the results broadly applicable.

The main contributions are two upper‑bound theorems that improve on earlier, looser estimates.

Theorem 1 (Vertex‑fault diameter).
Assume that (F) is (k_{F})-connected and (B) is (k_{B})-connected, with (0<a<k_{F}) and (0<b<k_{B}). If the additional conditions
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